J. F. Sharpey-Schafer

Physics Department, University of Western Cape, Belleville, South Africa.

Collective Excitations of Deformed Nuclei and their Coupling to Single Particle

States

Bohr & Mottelson Vol II

Page 363 !!

Self-consistent Relativistic Mean Field +BCS

Li et al. PR C79(2009) 054301Nikšić et al.

PRL 99 (2007) 092502

15464Gd90

JFS-S et al. EPJ A47 (2011) 5

Congruence of first & second Vacuum structures in 154Gd & 152Sm

Configuration Dependentor Quadrupole Pairing

R. E. Griffin, A. D. Jackson and A. B. Volkov, Phys. Lett. 36B, 281 (1971). Suggested that Δpp ≈ Δoo >> Δop

for Actnide Nuclei where 02+ states were observed in (p,t) that

were not pairing- or β-vibrations.

W. I. van Rij and S. H. Kahana, Phys. Rev. Lett. 28, 50 (1972).

S. K. Abdulvagabova, S. P. Ivanova and N. I. Pyatov, Phys. Lett. 38B, 251 (1972).

D. R. Bès, R. A. Broglia and B. Nilsson, Phys. Lett. 40B, 338 (1972).

took up the suggestion

I. Ragnarsson and R. A. Broglia, Nucl. Phys. A263, 315 (1976).

coined the term “pairing isomers” for these 0+ states

Single-Particle

Quadrupole Moments

in a deformed W-S

potential

A=155

A=239

N=90[505]11/2-

Abdulvagabova, Ivanova & Pyatov

Phys. Lett. 28B (1972) 215

Low Density of Oblate s-p States Below theFermi Surface

[505]11/2-

“Oblate”

[660]1/2+

“Prolate”

[521]3/2-90

Nilsson Diagram N~90Configuration Dependent or QuadrupolePairing; Assume Δpp≈ Δoo>> Δop

What is theWhat is the │0│022+ + >> Configuration ? Configuration ?

۞ (t,p) & (p,t) │02+ > is 2pn- 2hn

this gives Jπ but nothingnothing on the orbit.

۞ Single particle transfer would give ln but does not populate │02

+ >.

In { │02+ > + neutron }, look to see

which orbit does NOTNOT couple to │02+ >.

CanadianJournalof Physics51 (1973)1369

McMaster

Løvhøiden,Burke &Waddington

157Gd(p,t)155GdGsb β

[505]11/2-

Kπ=15/2-

=2γ+ + [505]11/2-

Missing │02+ >

coupling

155155GdGd9191

High-K High-K Decay Decay SchemeScheme

1000

1200

1400

800

600

400

200

0

Ex

(keV)

[521]3/2-

[651]3/2+ [505]11/2-

│02+ >

BLOCKEDBLOCKED

155Gd91

Seen by Schmidt et al;

J. Phys. G12(1986)411

in (n,γ) (d,p) & (d,t)

[642]5/2+[402]5/2+

[532]3/2-

[400]1/2+

681 keV

K=3/2-

K=3/2+ {K=11/2-}

Kγ=2

K=1/2-

=2-Ω

K=1/2+

=2-Ω K=15/2-

=2+Ω

996 keV

Gamma Phonon Energy (keV)

600

700

800

900

1000

1100

1200

1300

0 100 200 300 400

E(2+;gsb) keV

E(P

ho

no

n)

keV N=88

N=90N=92N=94N=96N=98

Egnd state = ½ ħωβ + ħωγ

Ex (0,0,2,2) = ħωγ + ħ2/I K = 2 Gamma Vibration Band Head Energy

Systematics of the strength of the E2 transitions from the 2γ+ γ-bandhead to the ground state in a series of deformed nuclei.

M1s

1. ΔK=2 kills the M1 component in J → J out-of-band transitions. Hence not much p-h component in the 2γ

+ states unless all the components manage to cancel out ??

2. In-band M1 transitions are very weak, demonstrating that gK ≈ gR

The way the Soloviev qp-phonon model (QPM) works is as follows; 1] He postulates that collective phonons exist. He does not know how or what.V. G. Soloviev, Nucl. Phys. 69, 1-36 (1965)

2] So he defines a phonon operator; Qi Ψ = 0 (his equ 6)And collective states are given by; Qi+ Ψ (his equ 7)And [Qi,Qj+] = δij (his equ 8)

3] As he has no clue at all what the Qi are, he assumes they can be expanded in terms of 2qp wave functions. That is in terms of particle hole states and nothing more complex.

4] To do this he needs an interaction which he assumes is of the multipole-multipole type.

5] The phonon energy he then finds using the variation principle for Qi+ Ψ (his equ 10). But to do this he has to fit experimental data as his Hamiltonian has unknown constants κn , κp , κnp (his equ 3) which govern the strengths of the neutron-neutron, proton-proton and neutron-proton interactions. Later he gets fed up and puts all these κ s equal !!

6] In reference; V. G. Soloviev and N. Yu. Shirikova, Z. Phys. A301, 263-269 (1981) , he concludes “that the two-phonon states cannot exist in deformed nuclei” (his abstract). This is because anti-symmetrization and the Pauli principle pushes up the two-phonon energy to 2.5 times twice the phonon energy in his model. At these energies the two-phonon states are well above the pairing gap and will get mixed to hell !!!

Two Neutron Transfer to 154Gd (N=90)

N.B. Log10 scale

Shahabuddin et al; NP A340 (1980) 109

Kπ

= 2

+ B

andh

ead

O. P. Jolly; PhD thesis (1976), McMaster+ Denis Burke & Jim Waddington

Proton Stripping to 154Gd

NB; Log10

Scale

0 2 4K=0+ │02+ >

Kπ = 2+ band

Spin I ( ħ )

K=2γ-band

groundstate band

aligned i13/2 band

156Dy from the148Nd(12C,4n)156Dyreaction

2nd

vacuum

Gammasphere Data Nov. 2008

Siyabonga Majola et al., to be published

Spin I ( ħ )

groundstate band

aligned i13/2 band

156Dy from the148Nd(12C,4n)156Dyreaction

K=2 γ-bandbuilt on thealigned i13/2

band

Some even-even nuclei in which the γ-band has been observed above 15+

Nucleus Beam Highest Spin Reached Reference

Species Energy (MeV) Yrast band γ-even γ-odd

104Mo ff (fission fragment) 20+ 18+ 17+ [49]

154Gd α 45 24+ 16+ 17+ [3]

156Dy 12C 65 32+ 28+ 27+ [50]

156Er 48Ca 215 26+ 26+ 15+ [51]

160Er 48Ca 215 50+ - 43+ [52]

162Dy 118Sn 780 Coulex 24+ 18+ 17+ [53]

164Dy 118Sn 780 Coulex. 22+ 18+ 11+ [53]

164Er 9Be 59 24+ 14+ 19+ [54]

164Er 18O 70 24+ 18+ 21+ [55,56]

170Er 238U 1358 Coulex. 26+ 18+ 19+ [57]

180Hf 136Xe 750 Coulex. 18+ 16+ 13+ [58]

238U 209Bi 1130 & 1330 Coulex 30+ 26+ 27+ [59]

238U146

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30

Spin I

Ex

- E

r (k

eV)

gsb

K=2+ 238U146

γ-band

gsb

Ward et al., Nucl Phys A600 (1996) 88

Coulex; 209Bi beam on thick target

-200

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20

Spin I

Ex

- E

r (k

eV)

gsb [523]7/2-K< 3/2-K> 11/2-

16567Ho98

Coulex

Thick

target

209Bi beam

5.4 Mev/u

Chalk River

8π

G Gervais, Dave Radford et al. Nucl Phys A624 (1997) 257

K<

K> gsb

Odd Proton Coupled to core γ-Vibration

γ=2+

gsb=7/2-K>=11/2-

K<=3/2-

104Mo62

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 2 4 6 8 10 12 14 16 18 20 22

Spin I

Ex

- E

r (k

eV)

gsb

2+g

4+gg

10442Mo62

105Mo63

-500

0

500

1000

1500

2000

2500

0 2 4 6 8 10 12 14 16 18 20 22

Spin I

Ex

- E

r (k

eV)

gsb

2+g

4+gg

10542Mo63

103Nb62

-500

0

500

1000

1500

2000

0 2 4 6 8 10 12 14

Spin I

gsb

2+g

4+gg

10341Nb62

Jian-Guo Wang et al

Nucl Phys A834 (2010) 94c

Two Phonon Excitations ??

116Cd(48Ca,6n)156Er

215 MeV

J M Rees, E S Paul et al

Phys. Rev. C83 (2011) 044314

Z = 68 N = 88

Gammasphere

Data, ANL

16068Er92

Odd Spin γ-band in 160Er

Gammasphere data; Ollier et al., Phys. Rev. C83 (2011) 044309

164Er96

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30

Spin I

Ex

- E

r (k

eV)

gsb

gamma band

S-band

16468Er96

γ-band

S-band

gsb

Steve Yates et al., PR C21 (1980) 2366

150Nd(18O,4n)164Er and Coulex with 136Xe

Steve Yates et al., PR C21 (1980) 2366

150Nd(18O,4n)164Er and Coulex with 136Xe

16468Er96

Summary of Experimental Data onSummary of Experimental Data on γγ-Bands-Bands

1. B(E2) decays out-of-band to ground state band;ΔK = 2 hence no M1 strength. Mixing ratios δ >>1

2. M1 & g-factors; in-band transitions ΔJ = 1 are very weak or zero.

3. Transfer reactions (pick-up & stripping) zero or very weak. Rare exceptions ?

4. Some bands Signature split, others not.

Connected to γ-deformation ??

5. Alignments; ALL γ-bands track their intrinsic configuration.

6. K>/K< splitting in odd nuclei; very few examples

TheoryTheory ?? ??

1. Rotation-Vibration Model; based on Bohr Hamiltionan

2. Phonon/Boson Models; [Bés, Soloviev, Piepenbring…….]

Start with Nilsson potential + BCS => Quasi-particles

Assume Phonons/Bosons exist and then invent an interaction to

produce them !! Then fiddle with the interaction and the truncation.

3. IBA, X(5) and other fairy tales !

4. RPA etc, more sucessful ?

5. Triaxial Projected Shell Model (TPSM) [Hara, Sun, Shiekh et al]

Hopeful ??

CONCLUSIONS ??CONCLUSIONS ??

1. More data at low and high spins might help ?

2. Coulex is a good way of connecting to real collective structures and keeps the spectra less complicated.

3. Are Kπ = 2+ “γ-vibrational” bands just a projection of the Zero Point Motion on the symmetry axis or are they more of a traditional Boson/Phonon ??

4. Do RPA or TPSM help ?

5. Non-Microscopic Models are no use at all. [IBA, X(5)…..]

6. Unlike phantom β-vibrations, γ-vibrations are a REAL collective motion !!

Many thanks to all my colleagues

S M MullinsR A BarkE A LawrieJ J LawrieJ KauF KomatiP MaineS H T MurrayN J NcapayiP VymersP Papka iThemba LABS +

Stellenbosch Univ.

S P Bvumbi S N T Majola Univ. of WesternJFS-S CapeT E MadibaD G Roux A Minkova Univ. of Sofia J Timár ATOMKI,

Debrecen

iThemba

LABS

Rhodes University

SOUTH AFRICASOUTH AFRICA U.S.A.U.S.A. CANADACANADA FRANCEFRANCE

L L RiedingerD L HartleyC BeausangM AlmondM P CarpenterC J ChiaraP E GarrettF G KondevW D Kulp IIIT LauritsenE A McCutchanM A RileyJ L WoodC H WuS ZuD CurienJ DudeckN Schunk